Mathematics for the interested outsider

Distributions

A vector field defines a one-dimensional subspace of at any point with : the subspace spanned by . If is everywhere nonzero, then it defines a one-dimensional subspace of each tangent space. A distribution generalizes this sort of thing to higher dimensions.

To this end, we define a -dimensional distribution on an -dimensional manifold to be a map , where is a -dimensional subspace of . Further, we require that this map be “smooth”, in the sense that for any there exists some neighborhood of and vector fields such that the vectors span for each .

Notice here that the don’t have to work for the whole manifold . Indeed, we will see that in many cases there are no everywhere-nonzero vector fields on a manifold . But over a small patch we might more easily find vector fields that are linearly independent at each point, and thus define a smooth -dimensional distribution over . Then more general smooth distributions come from patching these sorts of smooth distributions together.

A vector field on “belongs to” a distribution — which we write — if for all . We say that is “integrable” if for all and belonging to .

Every one-dimensional manifold is integrable. To see this, we note that if and belong to then for some constant , at least at those points where . Thus we see that

and so is proportional to , and thus belongs to . To handle points where , we can put the scalar multiplier on the other side.

I think my difficulty comes from confusion over when juxtaposition denotes multiplication in the ring, multiplication of the ring over the module, function application, function composition, or even something else. (I had similar difficulties with representation theory :-) In a term like “fXYg”, the empty spaces between the letters can mean different things depending on how it’s parenthesized; I still haven’t wrapped my head around whether all the different ways make sense and that (among the ones that do make sense — all of them?) they all mean the same thing.

It’s possible to fully-parenthesize these sorts of expressions, but then they get confusing for a different reason. In general, I try to write things out so they associate to the right, because of function application, and use parentheses for particularly ambiguous setups.

In , we could write , but we’ll conventionally drop the parens and write for the application of the vector field to the function. There’s no way of “multiplying” vector fields, so has to be interpreted as . And it doesn’t matter whether we multiply by before applying it to or after, since and give the same result.

About this weblog

This is mainly an expository blath, with occasional high-level excursions, humorous observations, rants, and musings. The main-line exposition should be accessible to the “Generally Interested Lay Audience”, as long as you trace the links back towards the basics. Check the sidebar for specific topics (under “Categories”).

I’m in the process of tweaking some aspects of the site to make it easier to refer back to older topics, so try to make the best of it for now.